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<p>Multiplying Eqs. <a href="" class="xref" data-knowl="./knowl/Xsol.html" title="Equation 7.7.1">(7.7.1)</a> and <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "Tsol" missing or not unique]</code> together, we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Xsol.html       ">
\begin{equation*}
u_n= X_n(x)T_n(t),
\end{equation*}
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<p class="continuation">satisfy the PDE <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "heatpde" missing or not unique]</code> and the BCs <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "heatbc" missing or not unique]</code> for each <span class="process-math">\(n=1,2,\cdots\text{.}\)</span> The functions <span class="process-math">\(u_n\)</span> are sometimes called the <dfn class="terminology">fundamental solutions</dfn> to the heat conduction problem <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "heatpde" missing or not unique]</code>,<code class="code-inline tex2jax_ignore">[cross-reference to target(s) "heatic" missing or not unique]</code> and <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "heatbc" missing or not unique]</code>. Then the formal solution is given by</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Xsol.html       ">
\begin{equation*}
u(x,t)=\sum_{n=1}^{\infty}u_n(x,t)=\sum_{n=1}^{\infty}C_n e^{-n^2\pi^2\alpha^2 t/L^2}\sin(n\pi x/L).
\end{equation*}
</div>
<p class="continuation">It remains only to satisfy the initial condition <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "heatic" missing or not unique]</code>.</p>
<span class="incontext"><a href="sec7_7.html#p-407" class="internal">in-context</a></span>
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